Non-planarity of SL(2,Z)-orbits of origamis in H(2)

We consider the (Formula presented.) -orbits of primitive (Formula presented.) -squared origamis in the stratum (Formula presented.). In particular, we consider the 4-valent graphs obtained from the action of (Formula presented.) with respect to a generating set of size two. We prove that, apart from the orbit for (Formula presented.) and one of the orbits for (Formula presented.), all of the obtained graphs are non-planar. Specifically, in each of the graphs we exhibit a (Formula presented.) minor, where (Formula presented.) is the complete bipartite graph on two sets of three vertices.